Alignment of Quasar Polarizations on Large Scales Explained by Warped Cosmic Strings. PART II: The Second Order Contribution

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1 Jounal of Modn Physics, 07, 8, ISSN Onlin: 53-0X ISSN Pint: Alignmnt of Quasa Polaizations on Lag Scals Explaind by Wapd Cosmic Stings. PART II: Th Scond Od Contibution Rinoud Jan Slagt, Asfyon, Bussum, Th Nthlands Amstdam Univsity, Amstdam, Th Nthlands How to cit this pap: Slagt, R.J. (07) Alignmnt of Quasa Polaizations on Lag Scals Explaind by Wapd Cosmic Stings. PART II: Th Scond Od Contibution. Jounal of Modn Physics, 8, Rcivd: Januay 7, 07 Accptd: Fbuay, 07 Publishd: Fbuay 7, 07 Copyight 07 by autho and Scintific Rsach Publishing Inc. This wok is licnsd und th Cativ Commons Attibution Intnational Licns (CC BY.0). Opn Accss Abstact W find an azimuthal-angl dpndnt appoximat wav lik solution to scond od on a wapd fiv-dimnsional manifold with a slf-gavitating U() scala gaug fild (cosmic sting) on th ban using th multipl-scal mthod. Th spctum of th sval ods of appoximation show maxima of th ngy distibution dpndnt on th azimuthal-angl and th winding numbs n of th subsqunt ods of scala fild. This bakup of th quantizd flux quanta dos not lad to instability of th asymptotic wavlik solution, du to th suppssion of th n-dpndncy in th ngy momntum tnso componnts by th wap facto. This ffct is tiggd by th contibution of th fiv dimnsional Wyl tnso on th ban. This contibution can b undstood as dak ngy and can tigg th slf-acclation of th univs without th nd of a cosmological constant. Th is a stiking lation btwn th symmty baking of th Higgs fild dscibd by th winding numb and th SO() baking of th axially symmtic configuation into a disct subgoup of otations about 80. Th disct squnc of non-axially symmtic dviations, canclld by th mission of gavitational wavs in od to sto th SO() symmty, tiggs th pssu T zz fo disct valus of th azimuthal-angl. Th can b a possibl lation btwn th cntly discovd angl-pfncs of polaization axs of quasas on lag scals and ou thotical pdictd angl-dpndncy and can b an vidnc fo th xistnc of cosmic stings. Th discovy of th incas of polaization at in small subgoups of th sval lagquasa goups (LQGs), th d shift dpndncy and th lativ ointation of th spin axs with spct to th majo axs of thi host LQGs, point at a factional azimuthal stuctu, w also found in ou cosmic sting modl. This pculia discontinuous lag scal stuctu, i.., polaizations dictions of multipls of, fo xampl, π o π, can b xplaind by th spctum of DOI: 0.36/jmp Fbuay 7, 07

2 R. J. Slagt azimuthal-angl dpndnt wavlik mods without th nd of convntional dnsity ptubations in standad D cosmological modls. Cafully compaison of th spctum of xtmal valus of th fist and scond od ϕ - dpndncy and th distibution of th alignmnt of th quasa polaizations is ncssay. This can b accomplishd whn mo obsvational data bcom availabl. Kywods Quasa Polaization, Cosmic Stings, Wapd Ban Wold Modls, U() Scala-Gaug Fild, Multipl-Scal Analysis. Intoduction Gnal lativity thoy (GRT) is by fa th most succssful thoy constuctd by thotical physicists. Its pdictiv pow is impssiv. Famous mpiical confimd xampls a th K black hol and th mission of gavitational wavs by mging black hols. Th a, howv, pdictd phnomna not yt dtctd by obsvations. An xampl is th axially symmtic (spinning) compact objct, i.., th Papaptou o Lwis-van Stockum solution. Anoth wll studid objct is th slf-gavitating cosmic sting solution. Fo an ovviw, s th wok of Vilnkin and Shllad [] and Andson []. Cosmic stings a U() scala gaug votx solutions in gnal lativity in th famwok of GUT s. This U() scala gaug fild with a Mxican hat potntial has livd up its putation in th thoy of supconductivity, wh votx lins occu as topological dfcts and in th standad modl of paticl physics. In cosmology it could tigg th inflationay piod of xpansion and could solv th hoizon and flatnss poblm. It cam as a supis that th lativistic sting-lik votx solution of Nilsn and Olsn [3] can b found in GRT [] [5]. Howv, gnal lativistic cosmic sting is still not found dictly o indictly by obsvations. Th intst in cosmic stings fadd away whn on found inconsistncis with th pow spctum of th CMB: Cosmic stings cannot povid satisfactoy xplanation fo th magnitud of th initial dnsity ptubations fom which galaxis and clusts gw. It tuns out that th upp bound of th mass p unit lngth Gµ 0 6. Futh, th spcial pattn of a lnsing ffct of cosmic stings is not found yt. Studis of th adiativ ffcts of cosmic stings mbddd in a FLRW spactim show that cylindical gavitational adiation is apidly dampd and is ngligibl in any physical gim [6]. Ths sting-cosmology spactims ssntially look lik a scald vsion of a sting in a vacuum spactim. Th is, howv, anoth possibility to tst th xistnc of cosmic stings. Th cntly discovd alignmnt of quasa polaizations on vy lag scals [7] [8] could b xplaind by considing cosmic stings on a wapd ban wold spactim [9] [0]. It was alizd cntly that cosmic stings could b poducd within th famwok of supsting thoy inspid cosmological modls and a vival of cosmic stings occud. Ths so-calld 6

3 R. J. Slagt cosmic supstings can play th ol of cosmic stings in th famwok of sting thoy o M-thoy, i.., ban wold modls. Supsymmtic GUT s can vn dmand th xistnc of cosmic stings. Sup-massiv stings with an ngy dnsity of Gµ a intsting [] bcaus thi gavitational impact will b much stong than GUT stings. Thy could b poducd whn th univs undwnt phas tansitions at ngis much high than th GUT scal. Ban wold cosmological modls w fist poposd by Akani-Hamd, Dimopoulos and Dvali (ADD) [] [3] and Antoniadis t al. [] and xtndd by Randall and Sundum (RS) [5] [6]. In ths modls, th xta dimnsion can b vy lag compad to th ons pdictd in sting thoy, i.., of od of millimts. Th diffnc with th standad supsting modl is that th compactification ly on th cuvatu of th bulk. Th hug discpancy btwn 3 th lcto-wak scal, M EW = 0 GV and th gavitational mass scal, 9 M Pl = 0 GV (hiachy poblm) will b suppssd by th volum of th xta dimnsion, o th cuvatu in that gion. This ffct can also b achivd in th RS modls by a wap facto. Th waknss of gavity in ths modls a fundamntal and th Planck ngy could b of ( TEV) that may b accssibl by LHC. It is possibl that ffctiv D Kaluza-Klin (KK)-mods a obtaind fom th ptubativ 5D gaviton. Ths KK-mods will b massiv fom th ban viwpoint. Futh, on usually consids a fin-tuning btwn th tnsion on th ban (D cosmological constant) and th 5D tnsion in od to nsu a zo ffctiv cosmological constant. Fo an ovviw of ban wold modls, s, fo xampl, Roy [7] [8] [9] and Shiomizu t al. [0] and fncs thin. Cosmic stings could hav tmndous mass in th bulk, whil thi wapd manifstations in th ban show consistncy with th 6 obsvd bound of Gµ 0 by th wap facto [], vn if its valu was at th Planck scal. Wavlik distubancs tiggd by th hug mass of th cosmic sting in th bulk, could hav indd obsvational ffcts in th ban. Evidnc of ths objcts would giv us infomation at vy high ngis in th aly stags of th univs. Mayb thy might actually povid us th bst obsvational window upon fundamntal sting thoy. On ason fo this conjctu is th possibility of th xtnsion of fundamntal stings into th bulk in wapd spactims. In this sach w xtnd ou pvious wok on th lation of th alignmnt of quasa polaizations with wapd cosmic stings [0]. Rcnt sach in this fild [] shows pculia colation of quasa polaization ointations in LQGs. Th alignmnts can b paalll o ppndicula to ach oth. Ev oth disctizations a not xcludd. W shall s that this bhaviou fits in ou ban wold modl. Howv, th a som pculia issus to b addssd. A ntwok of long cosmic stings can b chaactisd by a singl scaling lngth, th psistnc lngth of th int-sting distanc. Numical simulations hav shown that th volution of th ntwok must b scal-invaiant. It is a gat challng to pov that th is a compaabl scal-invaianc in th alignmnts of th polaization vctos of quasas. By th discovy of high-dshift supmassiv black hols, z > 6, on could xtnd th invstigation on quasa 65

4 R. J. Slagt alignmnt to this a. Th is alady a tiny indication of z-dpndncy in th obsvd quasa polaization ointations [7]. Futh, th fomation of small- scal closd loops of cosmic stings can distub th ngy balanc of long stings. In ou modl this poblm is ovcom by th wap facto: it maks th cosmic sting mo massiv duing th volution. In Sction w outlin th multipl-scal mthod on a wapd ban wold spactim. In Sction 3 w calculat th mtic ptubations to scond od. In Sction w div th matt fild quations to scond od and indicat th possibl lation with axially symmtic instabilitis causd by adiationactions. In Sction 5 w discuss th possibl connction of th wap facto with confomal invaianc. In th appndics w collctd all th lvant quations in od to kp th main txt adabl and slf-containd.. Th Multipl-Scal Appoximation on a Wapd Ban Wold Spactim W will invstigat votx-lik solution on a wapd fiv-dimnsional Fidmann-Lmat-Robtson-Walk (FLRW) modl in cylindical coodinats [9] [0]: ( ) ( ) () γ ψ ψ ds = dt d dz ψ dϕ d y, with a wapfacto dpndnt of, t and th bulk dimnsion y. Th slfgavitating scala-gaug fild, paamtizd as inϕ Φ= ηx( t, ), Aµ = Pt (, ) n µ ϕ, () sids on th ban. η is th vacuum xpctation valu of th scala fild, n th winding numb and th gaug coupling constant. Th winding numb (numb of jumps in phas of th scala fild whn on gos aound th flux tub) is latd to th quantizd flux in th Ginsbg Landau thoy of supconductivity (Abikosov votics) and th disct valus of th topological chag in th sin-godon thoy. Th xact solution of [9] follows fom th 5D Einstin quation 5 κδ 5 ( )( ), Λ th ban tnsion and x ( tx, i, y) G = Λ g y Λ g T (3) with κ 5 = 8π G = 8π Mpl, µ =. Th 5 M pl is th fundamntal 5D Planck mass. Th scala-gaug fild quations bcom [] µ dv µ * * D Dµ Φ=, F * νµ = i ( Φ( Dν Φ) Φ Dν Φ), () dφ with Dµ Φ µ Φ ia µ Φ, µ th covaiant divativ with spct to g and V th potntial of th Ablian Higgs modl. Th sta psnts th complx conjugatd. F is th Maxwll tnso. Th modifid Einstin quations bcom [0] G = Λ ff g κ T κ5, (5) 66

5 R. J. Slagt with G th Einstin tnso calculatd on th ban mtic g 5 = g nn µ ν and n µ th unit vcto nomal to th ban. W will consid h Λ ff = 0, so w a daling with th RS-fin tuning condition [5]. Th last two tms on th ighthand sid of Equation (5) psnt th quadatic contibution of th ngy-momntum tnso and th lctic pat of th fiv dimnsional Wyl tnso spctivly. All standad modl filds a bound to th ban, although gavity may popagat into th xta dimnsion. Is is obvious, that th cosmic sting can build up a hug mass Gµ by th wap facto and can induc massiv KK-mods flt on th ban. Th wap facto causs ptubations to b dampd as thy mov away fom th ban, so gavity looks fou dimnsional, at last ptubativly, to a ban wold obsv. Ban wold modls can also xplain th acclation of th univs without th nd of a cosmological constant [8]. Distubancs can suviv th natual damping by xpansion of th univs du to th wap facto. This ffct was also found numically [9]. H w will consid th modifid cosmic sting fatus on th wapd spactim Equation () and us th multipl-scal appoximation [3] [] [5] in od to find patial diffntial quations (PDE s) fo th ptubations to scond od of th mtic and matt filds. This xtnds ou pvious sach [0]. Lt us xpand th mtic fild and th scala-gaug filds in th multipl-scal schm g = g ( x) h ( x, ξ, χ, ) k ( x, ξ, χ, ), ω ω Aµ = Aµ ( x) Bµ ( x, ξ, χ, ) C µ ( x, ξ, χ, ), (6) ω ω Φ=Φ ( x) Ψ ( x, ξ, χ, ) Ξ ( ( x, ξ, χ, ), ω ω with g th backgound mtic and Φ, A µ th backgound scala and gaug filds. Fo th scala fild w tak diffnt winding numbs, so diffnt magntic flux quantization fo th backgound fild and high od pinϕ inϕ tubations. W dfin Φ= ηx( t, ), Ψ= Y( t,, ξ) and 3 Z( t,, ) in ϕ Ξ= ξ. So w bak up th oiginal votx with winding numb n in ou cas in th stings with winding numbs n, n and n 3. On can pov [6] that this bakup mains stabl if th gaug to scala mass is >. In ou cas stability will b guaantd by invs pows of th wap facto. Futh, w paamtiz Bµ = [ B0, B0,0, B,0 ], Cµ = [ C0, C0,0, C,0], which will fulfil th highst od ptubation quation of th gaug fild as w shall s. Rapid Θ ptubations occu in th diction of th wav vcto lµ. W dfin µ x dg g g = g, l g g, g σ ω σ σ, (7) σ σ dx x ξ Substituting th xpansions into th ffctiv Einstin and matt fild quations, on obtains a st of PDE s fo g, h, k and th scala gaug filds Φ, Ψ, Ξ, A, B and C µ. Th ptubations can b ϕ -dpndnt. µ µ 67

6 R. J. Slagt Th Einstin quations in subsqunt ods of appoximation bcom ( ) ( ) ( ) ω G : =, (8) ( ) ( ) ( ) ( ) ( 0) ( 0) ( 0) 0 0 G G = T T 5 ω : κ κ, (9) ( ) ( ) ( ) ( ) ( ) : G = T 5. (0) ω κ κ W will consid h th quations up to od ( ) ω. W usd th notation fo th sval tms in th xpansion of a tnso, vcto o scala: ( ) ( 0) ( ) Vi = ωv Vi Vi Vi. Th contibution fom th bulk spac,, ω must b calculatd with th 5D Rimann tnso. If w consid ll µ µ = 0, i.., ( ) th Eikonal quation (which follows fom th ω scala quation), thn on obtains fom Equation (8) a st of stictions on h, such as th gaug α condition l h αν gαν h = 0. Lt us consid as a simplifid cas [,,0,0,0] l µ =. Thn w lt suviv h, h3, h, h and h 55 as indpndnt fist od ptubations of th mtic. 3. Th Mtic Ptubations up to Scond Od Th PDE s fo th backgound filds W,, ψ γ and th fist od ptubations h can b divd fom Equation (9)and w found in [0]. By intgating Equation (0) with spct to ξ, on obtains ( 0) ( 0) ( 0) ( 0) G = T 5 ( T 5 G ) κ κ κ κ d ξ, () τ with τ th piod of th apid vaiation. Th last two tms on th ight hand sid of Equation () can b intptd as back-action contibution of th KK-mods of th ptubativ 5D gaviton and can act as a cosmological constant. Ths quations can b usd to liminat th backgound filds fom th fist od PDE s. On thn obtains popagation quations fo th fist od ptubations. Fo compltnss w collctd thm in appndix A. Fom ths quations on obsvs that to fist od th is an intaction btwn th high-fquncy ptubations fom th bulk, th matt filds on th ban and th volution of h ij, also found in th numical solution [9]. Th bulk contibution h 55 is amplifid by W. It is a flction of th massiv KK mods flt on th ban. Th most intsting quation is th diffntial quation fo h, i.., th ( t, ϕ ) componnt, Equation (). It tiggs th ϕ - dpndnt distubancs. Th sin ( n n) ϕ -tm, amplifid by wap facto W, can hav xtmal valus on [ 0, π ], if w choos, fo xampl, n n =. W thn hav th tm cos ϕ, which has two xtmal valus on ( ) mod π (also found in [0]). Lt us now invstigat th high od quations in ω, i.., Equation (0), [ 0, π ] which will povid us fist od quations of t k and scond od q- 68

7 R. J. Slagt uations fo tt h. With th hlp of an algbaic manipulation pogam, on obtains, fo xampl, th quations fo k t and k t 55, w w took fo th momnt k3 = k3, k = k, k3 = 0 and k = ( k k ). S Appndix B. W obsv in Equation (33) again a cos ( n n) ϕ -tm amplifid by th wap facto. In th quation Equation (3) fo k t, th appas bsids th tm sin ( n n) ϕ, also a sin ( n ) 3 n ϕ -tm in connction with th scond od ptubation Z, amplifid by th wap facto. In th quation fo th fist od countpat quation, i.., Equation (), th is only th sin ( n n) ϕ -tm. So if w tak in Equation (3) fo ( n3 n) = (and ( n n) = ), thn th maxima in ϕ of ths two tms blonging to th ptubations of fist and scond od spctivly, a out-of-phas. In th nxt sction this will also bcom cla by considing th ngy-cunt componnts of th ngy momntum tnso. Fom Equation (33) and Equation (6) w can obtain a scond od PDE fo h 55 if w impos constaint conditions on k 55 o intgat Equation (33) with spct to ξ. This can also b don fo th oth componnts. So w can constuct with th multipl-scal mthod unifomly valid wavlik appoximations to solutions of ptubation poblms without without sonanc intactions btwn th conscutiv ods of ptubation thoy. This sult is latd to th Cauchy poblm. In any fild thoy wh th is a gaug fdom (as, fo xampl, in GRT and Maxwll thoy), on has to spcify gaug conditions in od to dtmin th dynamical volution fo som initial st of Cauchy data. In Maxwll thoy on usually choos th Lontz gaug. In GRT on has constaint quations bcaus th systm is ov dtmind. Ths constaints a usually PDE s of fist od. In ou appoximation schm w hav Equation (8) which lads to conditions on h. Th fist od quations Equations (3)-(7) can also b considd as constaint quations fo th scond od wav quations fo h, as xampl, in th cas of h 55. So w can constuct a dynamical volution of th systm of quations which fulfil th Cauchy data. W shall s in th nxt sctions how this sult can b applid in contxt with th cntly obsvd alignmnt of th polaization vctos of quasas ov lag distancs and th disctnss in th azimuthal dpndncy of th polaization axs.. Th Matt Fild Equations and th Engy-Momntum Tnso Expansion Fom Equation () w obtain, aft substituting th xpansions of Equation (6), ( ) fom th ω quations th conditions ll µ µ Ψ= 0 and l µ B = µ 0. So w will paamtiz B = µ [ B0, B0,0, B,0] and l µ l 0 µ =, othwis Ψ must b zo. Fom th fist od quation of th scala fild w obtain α * α ( 0) DDα Φ βφ( ΦΦ η ) = ( h ll µ ν Ψ g Γ lα Ψ ) d ξ, τ () wh w hav intgatd th quation with spct to ξ. On th ight hand 69

8 R. J. Slagt sid w s again th high-fquncy contibution to th fild quation. In ou cas, this back action tm tuns out to b zo. So th fist od quation is just th unptubd quation fo X. S Equation (8) of Appndix A. This also holds fo th quation fo A µ. S Equation (9) of appndix A. If w substitut back th intgatd quations into th oiginal quations, w thn α obtain th fist od ptubations (fo lc 0; Y( t,, ) in ϕ α = Ψ= ξ ). S Equation (30)-Equation (3) of Appndix A. Futh, w usd l α A = α 0, othwis th al and imaginay pats of Ψ intact as th popagation pogsss. Again, th appas a ϕ -dpndnt tm in th popagation quation fo B 0, amplifid by W. This dviation fom axially symmty was also found by [3]. Mo insight in this ϕ -dpndncy can b obtaind by studying th scond-od matt fild quations. On obtains fo Z and C again fist od diffntial quations. S Appndix B. W usd th fact that th complx conjugat of th full complx scond od scala quation also must b satisfid. Th appaanc of th tms cos ( n3 n) ϕ,cos ( n3 n) ϕ and cos ( n3 n n) ϕ will contibut to th nxt od mods of maxima in ϕ - dpndnt distubancs. W can obtain, as in th cas of th scond od mtic componnts, again scond od PDE s fo B and tt Y tt by suitabl constaints on C and Z o intgation with spct to ξ. Aft som aangmnt of Equation (35) fo xampl, w gt th wav quation fo Y (aft suitabl constaints on Z ). γ Y γ ψ tty = Y Y ( n n P) βw YX YW tytw Y ψ ψ tψ h W W Y W W ψ ψ γ γ h ψ γ t t t W W ψ γ ψ γ Y Y h ( t Y t Y ) h W W ψ γ Y ( th h ) cos ( n n ) W ϕ γ ψ βyw X cos ( n n) ϕ W tw Z tz Z cos ( n3 n) ϕ, W wh is an xpssion in backgounds filds, h and h. W hav again two piodic functions cos ( n n) ϕ and cos ( n ) n ϕ with fquncy diffnc of a facto two and wh on of th functions is amplifid by W. It will b ncssay to study ths quations numically in od to compa th amplituds of ths two piodic functions with thos of th azimuthal-angl dpndnt maxima of th quasa polaization alignmnt. Th scond od quation fo B 0 can b obtaind fom th sum of th t- and -componnts of th scond od gaug fild quations. (3) 70

9 R. J. Slagt W can calculat th th fist tms of th ngy momntum tnso T. ( ) S Appndix C. In T tt th appas, fo ( n n) = and ( n3 n) =, th tms cos( ϕ ), sin ( ϕ ) and cos( ϕ ), whil in th fist od tm, Equation (), th is only th cos( ϕ ). This is also tu fo th ngy- ( 0) ( ) cunt componnts Tt ϕ and Tt ϕ. In th ngy momntum tnso com- ( ) ponnt T tt th is also a tm popotional ( n n P). In th nxt od ( ) T tt th will b tms popotional with high ods of ( n n P). Ths high od tms will b suppssd by th wap facto, so th votx will not bcom unstabl as is th cas whn on bakup th votx sting in multipl flux [7]. Th most intsting bhavio aiss in th angula componnt T ϕϕ, i.., Equation (3)-Equation (5). As alady noticd by Laguna-Castillo and Matzn [5], T ϕϕ can altnat in sign dpndnt of th gaug to scala mass. This can ( 0) also happn dynamically [9]. In th nxt od T ϕϕ w hav th Y con- tibution in font of cos[ ϕ ] and in th nxt od in font of cos[ ϕ ] (fo th chosn valus of ( ) T ϕϕ th Z contibution n i as abov). So th doubling of ( 0) th fquncy in obvious. Fom th xpssion fo T zz, Equation (6), w s that th pssu in th z-diction is again dominatd by th cos ( n n) ϕ, bcaus th scond tm is suppssd by W. Th is, howv, a pculia sid ffct: th tm ( tx X) can chang sign dynamically. A numical solution can giv a dcisiv answ. Th is a lation btwn th phas fdom inϕ of ou scala fild and th scula instability of an initially quasi-stationay axially symmtic configuation causd by adiativ action. Th small non-axially symmtic dfomations tun out to b of th fom imϕ with m an intg [8]. This bokn symmty, dscibd by th invs of th angula momntum J, is compaabl with th symmty baking of th Higgs fild considd in ou modl. An axially symmtic systms is invaiant und otations in two dimnsions, th SO() goup. Th baking of this symmty can b xpssd in th quatoial ccnticity. Th paticula ointation of th llipsoid in th (x-y) fam can b xpssd though th azimuthal angl ϕ. This disct chang into non-axially symmty must b canclld by mission of gavitational ngy (and is am- plifid in ou modl by th 5D contibution), othwis w a saddld with a hlical tim coodinat, t t Jϕ and must giv up Lontz invaianc. This is cla fom th fact that ou mtic will thn possss a gt ϕ tm. Th angula momntum in (x-y) plan is dtmind by th cunts of th momntum ρ νµ ν ρµ dnsity, xt xt and can b calculatd in ou cas with th off-diagonal i 0 j j 0i J d x xt x T. componnts of T of Appndix C. Fo xampl ij ( ) 5. Quasa Polaization Alignmnt and Scal-Invaianc In od to xplain th cntly found lag-scal alignmnts of th polaization vctos of quasas in LQGs at cosmological dshifts z.5 by cosmic stings, it would b dsiabl to find a kind of scal (confomal)-invaianc, bcaus it is conjctud that cosmic sting ntwoks volv scal-invaiant just aft th 7

10 R. J. Slagt adiation dominatd a of ou univs. Th inticat fatus of th po- laization axs alignmnts of th quasas in LQGs show at last a kind of co- volution at vy lag scals [], so a study of confomal symmty of ou modl could b of intst. Gavity thoy invaiant und g ( x) ( x) g ( x) Ω is local confomal invaiant and must b spontanously bokn bcaus ou wold appas not to b scal invaiant [9] [30]. Lt us wit ou spactim of Equation () with ψ = γ = 0 ds =Ω dt d dz dϕ d y. Ω () wh w namd th wap facto as Ω. If w consid th ( t, ) -dpndnt pat of Ω and consid th flat (ban) cas of th mtic Equation (), ˆd d d d d s = t z ϕ d y. (5) Ω w thn obtain fo th Ricci scala ˆ Ω R = Ω Ω ( Ω Ω ). Ω Ω 5 tt t Th Ricci scala tansfoms und g ( x) ( x) g ( x) Ω as [3] gˆ Ω Ω ˆ ˆ 8gˆ Ω ˆ ˆ Ω Ω 5 ˆ 5 R Rˆ µ ν µ ν Ω 8 ( tt ) 5 Ω tω Ω Ω Ω ˆ = R. Ω Ω Ω (6) (7) So fo confomal invaiancy of 5 ˆR, th scond tm on th ight hand sid of Equation (7) must vanish. Fo 5 R ˆ = 0 w thn find Ω Ω Ω= 0, (8) tt with constaint quations tω Ω = 0. Equation (8) is just th quation of a vibating cicula dum. Th gnal solution fo th bounday conditions Ω,0 = f, Ω,0 = 0 is ( ) ( ) t ( ) ( ) ( ) ( ) ( )( 0( ) 0( )) Ω t, = Acos ct Bsin ct J c Y c, (9) n n n n n with J 0, Y 0 Bssl functions and n f. Ths solutions psnt fo suitabl bounday conditions, th standing nomal mods of th ban in th vacuum cas. In gnal, Ω can also dpnd on th azimuthal angl. This dpndncy is found, in ou non-vacuum situation, in th pcding sctions in th multipl-scal appoximation. So on could conclud that th wap facto in th vacuum cas fulfils a scala wav quations psnting fluctuations of th ban in th gound stat. It psnts th amount of local sttching of th D gomty. In th non-vacuum cas, with th U() C cofficints dpndnt of ( ) 7

11 R. J. Slagt scala gaug fild in th ban, on can ty to fomulat again th confomal invaianc. This is a pculia issu in thotical physics till now. Einstin quations and th scala quation (Klin-Godon quation)) a not confomally invaiant. On has to modify Einstin s quations to mak it confomally invaiant and mak th ngy momntum tnso taclss [3] [3]. Ou Ω - fild can play a cucial ol in this contxt if on intoducs an unavoidabl dilaton fild. 6. Conclusions It is found on a fiv dimnsional wapd ban wold spactim, using a multipl-scal appoximation schm, that to scond od th mtic and scala gaug fild show a spctum of azimuthal-angl dpndnt wavlik mods with xtmal valus dpndnt of th winding numbs of th backgound, fist and scond od ptubations of th scala fild. In fou dimnsional modls, this local fild thoy admits votx-lik bhavio and is a gnalization of th Ginzbug-Landau thoy of supconductivity. A lattic of Abikosov votics can b fomd, caying a quantizd flux dpndnt of th winding numb o votx chag n. Votics with n > a unlikly, sinc th ngy is ducd if thy split up into singl votics. Howv, in th gnal lativistic cas, gavity coms into play and th ngy of th configuation of th votics must b calculatd covaiantly by mans of th ngy momntum tnso. This gnal lativistic votx solution (cosmic sting) can build up a hug mass p unit lngth in th bulk and can induc massiv Kaluza-Klin mods flt on th ban, wh th standad modl filds sid. Distubancs don t fad away duing th xpansion of th univs du to th wap facto. Th jump in th phas of th scala fild is latd to th scula instability of th initially stationay axially symmtic configuation causd by th adiation action. Th baking of th axially symmty, dscibd by th invs of th angula momntum, is imϕ (m an intg and ϕ th azimuthal angl), compaabl with th symmty baking of th scala fild. Th covy of th SO() symmty fom th quatoial ccnticity is tiggd by th mission of gavitational wavs. Ou modl can b usd to xplain th mystious alignmnt of quasa axs with th lag-scal stuctu of ou univs and can sv as vidnc fo th xistnc of cosmic stings. Th found factional azimuthal stuctu is compaabl with th angl-pfncs of th polaization axs of quasas. Th is a stong vidnc of scaling (confomal) bhaviou of long nonintcommuting cosmic stings ntwoks duing th adiation-dominatd a. High o low initial sting dnsitis tnd towad a fixd scaling valu. Howv, standad cosmology, constaints such stings to b vy light and will fad away (o disappa by th foming of closd loops). In ou modl thy can suviv by th wap facto. This fact maks th compaison with th alignmnt of quasas possibl. So it would b dsiabl to hav also a scal-invaiant alignmnt 73

12 R. J. Slagt stuctu. This is cuntly und study. Rfncs [] Vilnkin, A. and Shllad, E.P.S. (99) Cosmic Stings and Oth Topological Dfcts. Cambidg Univsity pss, Cambidg, UK. [] Andson, M.R. (003) Th Mathmatical Thoy of Cosmic Stings. IoP Publishing, Bistol. [3] Nilsn, H.B. and Olsn, P. (973) Nucla Physics B, 6, 5. [] Gafinkl, D. (985) Physical Rviw D, 3, [5] Laguna-Castilo, P. and Matzn, R.A. (987) Physical Rviw D, 36, [6] Ggoy, R. (989) Physical Rviw D, 39, [7] Hutsmks, D., Baibant, L., Plgims, V. and Slus, D. (0) Astonomy & Astophysics, 57, A8. [8] Taylo, A.R. and Jagannathan, P. (06) Monthly Notics of th Royal Astonomical Socity, 59, L36. [9] Slagt, R.J. and Pan, S. (06) Foundations of Physics, 6, [0] Slagt, R.J. (06) Jounal of Modn Physics, 7, [] Laguna-Castillo, P. and Gafinkl, D. (989) Physical Rviw D, 0, [] Akani-Hamd, N., Dimopoulos, S. and Dvali, G. (99) Physics Ltts B, 9, [3] Akani-Hamd, N., Dimopoulos, S. and Dvali, G. (999) Physical Rviw D, 59, Aticl ID: [] Antoniadis, I., Akani-Hamd, N., Dimopoulos, S. and Dvali, G. (998) Physics Ltts B, 36, [5] Randall, L. and Sundum, R. (999) Physical Rviw Ltts, 83, [6] Randall, L. and Sundum, R. (999) Physical Rviw Ltts, 83, [7] Maatns, R. (007) Jounal of Physics: Confnc Sis, 68, Aticl ID: [8] Maatns, R. (007) Lctu Nots in Physics, 70, [9] Maatns, R. and Koyama, K. (00) Living Rviws in Rlativity, 3, 5. [0] Shiomizu, T., Mada, K. and Sasaki, M. (000) Physical Rviw D, 6, Aticl ID: 00. [] Slagt, R.J. (0) Intnational Jounal of Modn Physics D, 0, 37. [] Plgims, V. (06) AXiv:asto-ph/6005v. [3] Choqut-Buhat, Y. (969) Communications in Mathmatical Physics,, [] Choqut-Buhat, Y. (977) Gnal Rlativity and Gavitation, 8,

13 R. J. Slagt [5] Slagt, R.J. (986) Astophysical Jounal, 307, [6] Bogomol nyi, E. (976) Sovit Jounal of Nucla Physics,, 9-5. [7] Flsag, B. (987) Gomty, Paticls and Filds. Odns Univsity Pss, Odns. [8] Chandaskha, S. and Lbovitz, N.R. (973) Astophysical Jounal, 85, [9] T Hooft, G. (0) AXiv: g-qc/06675v3. [30] T Hooft, G. (05) AXiv: g-qc/507v. [3] Wald, R.W. (98) Gnal Rlativity. Th Univsity of Chicago Pss, Chicago. [3] Maldacna, J. (0) AXiv: g-qc/05563v. 75

14 R. J. Slagt Appndix A. Th Backgound and Fist Od Ptubation Equations In an ali wok [0] w obtaind th quations fo th backgound mtic componnts ( W,, ψ γ ), backgound matt filds XY, and fist od appoximation quations of h 3, h, h, h, Y, B and B 0. Thy a W W = W W W W ( ) ( ψ ψ ) ( γ γ ) tt t t t W ( )( ) W W ψ ψ γ γ W ψψ t t t t ( tp P) WtW 3 ψ tw κ W ( t X X), W W ψ W ψ = ψ ψ ψ ( W W ) tt t t W W ψ 3 κ W γ ψ ( tp P W X P ), W = ( ) tγ γ W tψ ψ tw W tw W W 3W WW t t W W tw W W W γ X P 7 tx 5 X X tx 5 t κ 6 ( P tp) ψ γ ψ 6 W β ( X η ). W ( )( ψ ψ) (0) () () tw W th 3 = h 3 k 3 k 3 tψ ψ h 3, (3) W h = h k k W W ψ ψ h t t t W γ γ ψ γ ψ κ W XPY sin ( n n ) ϕ ϕ W h 55 h h, γ th h ψ tψ h k k k ( ) = ( tw W W ( ψ tψ tγ γ )) h W γ ψ W tw W h 55 κ ( tx X) Y cos ( n n ) ϕ, W th55 h55 0, () (5) = (6) 3 tw W th = h ψ tψ h W κ ψ ( P tp) B W tψ ψ h 55. (7) 76

15 R. J. Slagt Th backgound matt filds ( X, P ) bcom [9] X W X t t tt X = X W W γ XP γ W ψ βx X η W X ( ), (8) P γ ψ tt P = P ( Pψ t Ptψ) W PX, (9) and ptubation quations W tw ty = Y Y (30) W ( tp P) ψ tb = B ψ tψ B h W ( tp P) γ ψ B = B B h t 0 0 ϕ W γ ψ YXW sin ϕ ( n n). Ths backgound quations don t contain ptubations tms, du to ou (simplifid) gaug conditions. B. Th Scond Od Ptubation Equations With th hlp of a algbaic manipulation pogam, w obtain fom Equation (0), fo xampl, th most intsting: tk 55 = k 55 ( h55 tth55 ) ( th55 h55 ) h 55 ψ γ γ ( th 55 h 55 ) h κ XY ( P ( n n P) W ( )) cos ( ) W W ( twtψ Wψ) ψ W h W W ψ γ ψ W β η X n n ϕ κ X P W ψ t 3 ψ ttψ ψ tψ W W h, (3) (3) ψ γ γ X P ψ κ β ( X η ) W W 8 tw W ttw W W h W W W W W W ψ ψ t ψ tψ ϕh ( h th ) h W W γ γ ϕh tϕ h ϕϕh h ϕϕ κ X PB, (33) 77

16 R. J. Slagt and k = k W W φ φ k h h h ( γ ψ t ) κ ( t ( ) ( ) t t t 55 t W h h55 h55 W XY n n P ( t )) sin ( ) ϕ sin ( 3 ) X P Y YB n n W XPZ n n ϕ ( tx X) Y ψ γ h cos ( n n ) ϕ ψ γ h γ ψ 55 ( ϕh W ϕh55 ) h ϕh W ψ γ γ ψ ( h h ϕ h ϕh ) ϕ ( W tψ WtW ) h W 55 γ tw γ ψ tγ tψ t 55 W h h k k k W k γ tw tψ h W ( ) γ ψ th W th55 γ γ ψ h h ϕ 55 h ϕh55 κw X P B0 βh ( X η ), 8 (3) with a function of th backgound filds and th filds h ij. Th oth componnts of th mtic ptubations a obtaind in th sam way. Th scond od quations fo Z and C a W tw γ ψ tz = Z Z βwxy cos ( n3 n n) W ϕ ψ γ Y YW tytw Y tty (( Y ty) h W W γ γ ψ Y ( th h )) Y ( n n P) β YX W ψ ψ γ h Y ψ tψ hy ( γ tγ ψ tψ W W W tw Y ty cos ( n3 n) ϕ W Y ψ γ ( tx X)( k k ) ( txth Xh ) W XW txtw h X tt X W ψ h ( txtψ X tψ Xγ tx tγ )) ( X ψ W γ γ (35) X XP tx tψ XPB cos ( n3 n) ϕ, 78

17 R. J. Slagt γ ψ tc = C ψ tψ C XWB cos ψ γ P W tw t B B tt P P B W W γ ψ ( B tt B tϕb0 ϕb0) XYW ( n n P) ( n n) P W PW h ψ t t W W ( tp tψ P ψ) ψ P B tψ ψ ( P tp) h h W (( tp P) k tp th P h B ( th h )) ψ W. ϕ (36) Th a th scond od patial divativ tms tty Y and B B in Equation (35) and Equation (36) spctivly. Thy can b tt isolatd in od to gt a wav quation fo th fist od ptubations. C. Th Engy Momntum Componnts Fo th sval ods of th ngy-momntum tnso componnts w find ( 0) tϕ ( ) = ( ) ( ) sin ( ) T XY n n P X P Y BY n n tϕ t t T ϕ = 0, (37) t ( ) ψ γ XPZsin n n Yh X X n n ( ) ϕ ( ) cos ( ) 3 t W T = XPY sin n n ϕ (38) ψ γ ψ γ X P X PB 0 β( X η ) h h ( tp P) X tx. 8 W W ϕ ϕ γ γ ψ ( ) ( ) β( η ) ψ T = P P X X X P W X tt W t t 8 ψ ( 0) = ( ) cos ( ) ϕ ( ( )) T Y Y X X n n B B P P tt t t W ( ) = ( ) cos ( ) ϕ cos ( ) T Z X X n n YZ n n ϕ tt t ψ 3 3 β XYB 0sin n n XY W X γ ψ ( ) ϕ ( η ) γ X Y txty Y ( n n P) cos ( n n) ϕ XY ( t ) ( t ) P P B P P B W X PW γ ψ h X X P h ψ 8 W β( η ) ψ γ Y( ty Y) C ( tp P B) XPB W (39) (0) () () 79

18 R. J. Slagt γ ( t ) ( t ) ψ γ Tϕϕ = P P X X W X P W β X 8 ( η ) ψ B T Y X X n n P P ( 0) γ ψ γ = ( t ) cos ( ) ϕ ( t ) ϕϕ W T W YX X X Y X Y ψ ϕϕ ( ) γ = β( η ) ( t t ) ψ γ γ XPY ( n n P) Y ( tx X )(( h h ) cos ( n n ) ϕ W γ ( Z ( tx X) ) cos ( n3 n ) ϕ βh ( X η ) 8 XPB ψ γ γ Y ( ty Y) C ( P tp) W ( h h ) h ( P P B( P P) ) ψ γ ψ γ γ t t W W. 6ψ γ ( 0) ψ γ Tzz = Y( t X X) cos ( n n ) ϕ B ( t P P). W (3) () (5) (6) Submit o commnd nxt manuscipt to SCIRP and w will povid bst svic fo you: Accpting p-submission inquiis though , Facbook, LinkdIn, Twitt, tc. A wid slction of jounals (inclusiv of 9 subjcts, mo than 00 jounals) Poviding -hou high-quality svic Us-findly onlin submission systm Fai and swift p-viw systm Efficint typstting and poofading pocdu Display of th sult of downloads and visits, as wll as th numb of citd aticls Maximum dissmination of you sach wok Submit you manuscipt at: O contact jmp@scip.og 80